2023-12-11 Q1(m=6)

Rearrange the digits in ⟨1263045⟩ to meet the rules below.

⟨6th 5th 4th 3rd 2nd 1st 0th⟩

✅Match

Jump(1,2) = 1

4th → 1

Jump(2,4) = 2

Jump(3,6) = 1

⛔Avoid

5th|4th|3rd|0th → 6

⟨   ⁵ᵗʰa       ¹ˢᵗb   ⟩, (ab)₁₀ ≥ 16

#125034_v2.2

       ┌───┬───┬───┬───┬───┬───┬───┐
       │6th│5th│4th│3rd│2nd│1st│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │   │   │ 1 │   │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 2 │   │ 0 │ 1 │   │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 3 │ 2 │ 0 │ 1 │   │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 4 │ 2 │ 0 │ 1 │ 4 │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 5 │ 2 │ 0 │ 1 │ 4 │   │ 5 │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 6 │ 2 │ 0 │ 1 │ 4 │ 6 │ 5 │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 7 │ 2 │ 0 │ 1 │ 4 │ 6 │ 5 │ 3 │▒
       └───┴───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

Proof of 2023-12-11 Q1(m=6)
═══════════════════════════

Notation: if nth -> a, then we write [nth] = a.

Plainly, our first step follows from ✅「4th → 1」. Next, to avoid ⛔「⟨   ⁵ᵗʰa       ¹ˢᵗb   ⟩, (ab)₁₀ ≥ 16」, we need [5th] <= 1, so [5th] = 0.

       ┌───┬───┬───┬───┬───┬───┬───┐
       │6th│ 5■│ 4■│3rd│2nd│1st│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │   │   │ 1 │   │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 2 │   │ 0 │ 1 │   │   │   │   │▒
       └───┴───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│   │ 2 │ 6 │ 3 │   │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

By ✅「Jump(1,2) = 1」, we have 2 = [6th] or [2rd]. The latter does not hold, for otherwise there is no way to match ✅「Jump(2,4) = 2」:

┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│ 2▲│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│   │ 0 │ 1 │   │ 2 │   │   │
└───┴───┴───┴───┴───┴───┴───┘

Consequently, we have 2 = [6th]. Then, using ✅「Jump(2,4) = 2」, we get

       ┌───┬───┬───┬───┬───┬───┬───┐
       │ 6■│5th│4th│ 3■│2nd│1st│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
       │   │ 0 │ 1 │   │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 3 │ 2 │ 0 │ 1 │   │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 4 │ 2 │ 0 │ 1 │ 4 │   │   │   │▒
       └───┴───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│   │   │ 6 │ 3 │   │   │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

Note that ✅「Jump(3,6) = 1」 implies {3,6} = {[2nd], [0th]}. Therefore, 5 = [1st].

       ┌───┬───┬───┬───┬───┬───┬───┐
       │6th│5th│4th│3rd│2nd│ 1■│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 4 │ 2 │ 0 │ 1 │ 4 │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 5 │ 2 │ 0 │ 1 │ 4 │   │ 5 │   │▒
       └───┴───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│   │   │ 6 │ 3 │   │   │   │
└───┴───┴───┴───┴───┴───┴───┘

Lastly, to avoid ⛔「5th|4th|3rd|0th → 6」, we finish by

       ┌───┬───┬───┬───┬───┬───┬───┐
       │6th│5th│4th│3rd│ 2■│1st│ 0■│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
       │ 2 │ 0 │ 1 │ 4 │   │ 5 │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 6 │ 2 │ 0 │ 1 │ 4 │ 6 │ 5 │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 7 │ 2 │ 0 │ 1 │ 4 │ 6 │ 5 │ 3 │▒
       └───┴───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│   │   │   │   │   │   │   │
└───┴───┴───┴───┴───┴───┴───┘

Q.E.D.

#125034_v2.2